There are implicit quantifiers in the statement “The square of any odd integer is of the form 8m + 1.”?
There are implicit quantifiers in the statement “The square
of any odd integer is of the form 8m + 1.” Write the statement with the
quantifiers explicit. The universe is the set of integers N. Use O(n) for
the sentence: “n is odd.” Caution: Make sure ALL variables are
quantified in your statement.
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I'm not good at formalising so I'll leave that to you, but this may help -
Odd integers are of the form, 2n + 1, where n is an integer.
On squaring, we have :
(2n + 1)^2 = 4n^2 + 4n + 1
Partially factor it :
= 4n(n + 1) + 1
Multiply the 4 by 2 and divide n(n + 1) by 2 :
= 8[n(n + 1)/2] + 1
Now, n and n + 1 are consecutive integers, one of which must be even.
Therefore, 2 will divide into n(n + 1) to obtain an integer - call it m.
Thus, the square of any odd integer is of the form 8m + 1.